Compressor CFD simulation method

developmentA CFD study

Johan Björk

Engineering Physics and Electrical Engineering, master's level

2018

Luleå University of Technology

Department of Computer Science, Electrical and Space Engineering

Compressor CFD simulation method

development:

A CFD study

Johan Bjorkjohbjr-2@student.ltu.se

Department of Engineering Sciences and Mathematics

Scania supervisor: Olle BodinExaminer LTU: Gunnar Hellstrom

June 26, 2018

Abstract

This master thesis project consisted of three parts that all were performedthrough CFD simulations with the purpose to develop Scania’s methods in thesubject of CFD. All parts included simulations on Scania’s SC92T70 centrifugalcompressor.

Part one consisted of performing a mesh study for the purpose of reliability,to investigate the convergence of different parameters by refining the boundarylayer. The method used is an inflation option called First layer thickness. Fivedifferent meshes were generated where the Richardson extrapolation methodwas used to examine the parameters between the mesh refinements.

From the result from the examined parameters, an approximate relative er-ror could be calculated to be less than 0.52 %, and a numerical uncertainty ofless than 0.35 %, between Mesh3 and Mesh4. In addition to that, Mesh3 hada simulation time of one hour less than for Mesh4. These results motivated theuse of mesh3 to be refined enough for further work in this thesis project. Thismesh ended at 37, 915, 257 number of elements.

The second part consisted of performing steady state CFD simulations, to ex-amine different parameters in order to find indications of the phenomena surge.Here, experimental data was used as reliance to perform CFD simulations onthe compressor. Design points from experimental data was used, that rangedfrom low mass flow rates where surge arises, to high mass flow rates where an-other phenomena called choke occur. Except for the design points taken fromexperimental data, a few extra design points where included at low mass flowrates (in the region of surge). The goal was that the analysis of the differentparameters would generate fluctuations on the result for the design points insurge region. Four different rotational speeds on the compressor were examined,56k, 69k, 87k and 110k revolutions per minute. A total of 140 different pa-rameters were examined, where 10 of these indicated on surge.

All of these parameters that indicated on surge where found in regions ofvicinity to the compressor wheel, which are the regions subjected to the phenom-ena. The parameters indicating on surge where mass flow, pressure coefficient,static pressure and temperature. Indications where found at the wheel inlet,ported shroud, and wheel outlet interfaces. The indications were only foundfor the two lower rotational speeds of the compressor wheel. To capture thebehaviour on higher rotational speeds, more design points in the region of surgeare needed, or transient simulations.

Part three of the thesis project consisted of investigating the methodology ofperforming a Conjugate Heat Transfer model (CHT) with the CFD code CFX.This part has not been performed by Scania before, so a big part of the problemwas to investigate if it actually was achievable. The goal was to use this modelto calculate the heat transfer between fluid and solid parts, as well as between

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the solid parts and the ambient. One question Scania wanted to answer wasif the CHT model could generate aerodynamic performance that correspondsto Scania’s traditional adiabatic model, as well as to experimental data of thecompressor.

In this part, both solid and fluid domains were included in the geometryto calculate heat transport, in contrast to the traditional adiabatic model thatonly uses the fluid domains. Because of that, a big part of the work consisted ofdefining all interfaces connecting together surfaces between all domains. Thisis needed to model heat transport between the domains. In the set up partin CFX, the CHT model differed a lot from the traditional adiabatic model inthat way that the outer walls was not set up as adiabatic anymore. In the CHTmodel, instead heat transfer is allowed between the outer walls of the fluids andthe solids.

From the result simulations, one could see that the CHT model was able tocompute the heat transfer between fluids and solids. It also managed to exportthermal data such as heat flux and wall heat transfer coefficient to be used formechanical analysis, which is an important part in Scania’s work. From the anal-ysis of aerodynamic performance, a conclusion was drawn that the CHT modelwas able to compute efficiency and pressure ratio that followed the behaviour ofthe traditional adiabatic model as well as experimental data. However, for lowermass flows, the CHT model started to underpredict which could be explainedby the geometrical differences between the CHT and adiabatic model.

By analysis of temperature, one could see quantitative differences comparedto the traditional adiabatic model. For other parameters (static and total pres-sure), there were no experimental data to be used for comparison. Because ofthat, an important part in future work of this CHT method development is toperform more experimental test for CFD data to be compared against. An-other important part to compare the models is to have an identical geometry.Without an identical geometry, deviations in result will occur that depends ongeometry.

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Sammanfattning

Detta examensarbete har bestatt i tre moment som alla har utforts genom CFD-berakningar med syfte att utveckla Scanias metoder inom CFD. Alla momenthar omfattat berakningar pa Scanias SC92T70 centrifugalkompressor.

Del ett bestod i att utfora en meshstudie for att i reliabilitetssyfte undersokaolika parametrars konvergens genom att forfina meshens gransskikt. Metodensom anvandes ar en inflationstyp som heter First layer thickness. Fem olikameshforfiningar togs fram dar olika pararametrar undersoktes med hjalp avRichardsons extrapolation.

Fran resultatet for de undersokta parametrarna kunde ett approximativt rel-ativt fel beraknas till att vara mindre an 0.52 %, och en numerisk osakerhet pamindre an 0.35 %, mellan Mesh3 och Mesh4. Mesh3 var dessutom en timmesnabbare att simulera pa. Med dessa resultat motiverades Mesh3 till att varatillrackligt forfinad kring gransskiktet for resterande arbeten inom detta exam-ensarbete. Mesh3 slutade pa 37, 915, 257 antal element.

Andra momentet bestod i att med hjalp av stationara CFD-berakningar, un-dersoka olika parametrar for att forsoka hitta olika indikationer pa surge. Dettamoment bestod av att anvanda experimentell data som stod, for att utforaCFD-simuleringar pa kompressorn. Har anvandes de designpunkter fran exper-imentell data som stracker sig fran laga massfloden dar surge uppstar, till hogamassfloden dar fenomenet choke ligger. Forutom designpunkterna tagna franexperimentell data lades aven nagra extra till, vid laga massfloden (i regionendar surge uppstar). Tanken var att analys av olika parametrar skulle ge utslag paresultatet for de inkluderade designpunkterna. Fyra olika rotatioinshastigheterpa kompressorn undersoktes, 56k, 69k, 87k och 110k varv per minut. Samman-lagt 140 olika parametrar undersoktes, och 10 av dessa gav indikationer pa surge.

Samtliga av dessa parametrar som indikerade pa surge fanns i omraden inarhet till kompressorhjulet, vilket ar den del av kompressorn som utsatts fordetta fenomen. Mer specifikt kunde parametrar som massflode, tryckkoeffi-cient, statiskt tryck och temperatur ge utslag. Indikationerna aterfanns pahjulinlopps-, ported shroud- och hjulutloppsgranssnittet. For de indikationersom fanns for fenomenet surge, sa kunde det bara sparas for de tva lagstaundersokta rotationshastigheterna. For att finna fenomenet for hogre rotation-shastigheter pa kompressorhjulet sa kravs flera designpunkter i omradet kringsurge, eller transienta simuleringar.

Del tre i detta examensarbete gick ut pa att undersoka metodologin foratt stalla upp en Conjugate Heat Transfer -modell (CHT) med hjalp av AnsysCFX. Detta har inte gjorts tidigare pa Scania sa stor del av problemet varatt undersoka om det faktiskt gick att utfora en CHT-modell. Tanken varatt anvanda denna modell for att berakna varmetransporten mellan fluid- ochsoliddelar, men ocksa mellan soliddelar och omgivning. En fragestallning som

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Scania ville ha besvarad var om modellen kunde generera aerodynamisk datasom motsvarar deras traditionella adiabatiska CFD-metod.

I detta moment anvandes bade solider och fluider i geometrin for att beraknavarmetransport, till skillnad mot den traditionella adiabatiska modellen som en-dast anvander fluiddelarna. Pa grund av det bestod en stor del av detta arbetei att definiera upp alla granssnitt som kopplar samman olika ytor. Detta for attkunna modellera varmetransport mellan delarna. I sjalva modelleringen skiljdesig denna modell fran Scanias traditionella modell pa sa satt att yttervaggarnainte modelleras som adiabatiska langre, utan nu tillater varmetransport.

Fran resultatet gick det att avlasa att modellen kunde berakna varmetrans-port mellan fluid- och soliddelar. Den klarade aven av att exportera data formekanisk analys, sasom hallfasthetsberakningar, vilket ar en viktig del i Sca-nias arbete. Vid analys av aerodynamisk prestanda sa kunde slutsatsen avCHT-modellen dras att den beraknar verkningsgrad, och tryckforhallande somhar ett liknande beteende som Scanias traditionella adiabatiska modell. CHT-modellen hade dock en viss underprediktering vid lagre massfloden som gick attspara till att geometriska skillnader mellan CHT-modellen och den traditionellaadiabatiska geometrin.

Vid analys av temperatur gick det att urskilja stora kvalitativa skillnaderjamfort med den traditionella metoden. For ovriga parametrar (statiskt tryckoch totaltryck) fanns ingen experimentell data att jamfora med. Darfor kravsdet for framtida metodutveckling av denna modell en mer omfattande experi-mentell analys, sa CFD-data kan evalueras. En annan viktig del for att kunnajamfora resultat mellan modellerna ar en identisk geometi. Utan det kan avvikelserforekomma som kan bero pa de geometriska skillnaderna.

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Acknowledgement

I would like to thank my examiner and supervisor Gunnar Hellstrom at thedivision of fluid and experimental mechanics at LTU for providing me with helpduring this thesis. His door has always been open both for this project butalso during many of the courses I have taken during my years at the university.My suvervisor Olle Bodin and co-worker Thomas Svensson at the departmentNMGD at Scania for taking up a big part of their time to assist me in my work.I also want to thank my family for supporting me and for always being therewhen I have needed it.

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Contents

1 Introduction 121.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.2 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . 121.3 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2 Theory 152.1 General Theory on centrifugal compressors . . . . . . . . . . . . 15

2.1.1 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . 162.1.2 Ported shroud . . . . . . . . . . . . . . . . . . . . . . . . . 172.1.3 Governing equations . . . . . . . . . . . . . . . . . . . . . 182.1.4 Reynolds-Average Navier-Stokes . . . . . . . . . . . . . . 192.1.5 Eddy-Viscosity modeling . . . . . . . . . . . . . . . . . . . 202.1.6 SST Turbulence model . . . . . . . . . . . . . . . . . . . . 212.1.7 Boundary layer . . . . . . . . . . . . . . . . . . . . . . . . 222.1.8 Performance . . . . . . . . . . . . . . . . . . . . . . . . . 232.1.9 Aerodynamics . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.2 Mesh dependency study . . . . . . . . . . . . . . . . . . . . . . . 282.3 Surge line prediction . . . . . . . . . . . . . . . . . . . . . . . . . 292.4 Conjugate Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . 32

3 Method 343.1 Mesh Independency study . . . . . . . . . . . . . . . . . . . . . . 34

3.1.1 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.1.2 Meshing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.2 Surge line prediction . . . . . . . . . . . . . . . . . . . . . . . . . 423.2.1 Boundary Conditions . . . . . . . . . . . . . . . . . . . . 48

3.3 Conjugate Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . 50

4 Results 544.1 Mesh independency study . . . . . . . . . . . . . . . . . . . . . . 544.2 Surge line prediction . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.2.1 Mass flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.2.2 Pressure recovery coefficient . . . . . . . . . . . . . . . . . 604.2.3 Static pressure . . . . . . . . . . . . . . . . . . . . . . . . 614.2.4 Temperature . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.3 Conjugate Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . 644.3.1 Temperature fields . . . . . . . . . . . . . . . . . . . . . . 644.3.2 Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . 664.3.3 Pressure Ratio . . . . . . . . . . . . . . . . . . . . . . . . 674.3.4 Temperature . . . . . . . . . . . . . . . . . . . . . . . . . 684.3.5 Static pressure . . . . . . . . . . . . . . . . . . . . . . . . 694.3.6 Total pressure . . . . . . . . . . . . . . . . . . . . . . . . . 70

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5 Analysis & Discussion 725.1 Mesh dependency study . . . . . . . . . . . . . . . . . . . . . . . 725.2 Surge line prediction . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.2.1 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . 735.3 Conjugate Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . 73

5.3.1 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . 76

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List of Figures

1 CFX module and workflow in Ansys. . . . . . . . . . . . . . . . . 132 The anatomy of a turbo. . . . . . . . . . . . . . . . . . . . . . . . 163 Illustrative schematic of the centrifugal compressor. Adapted

from Hill & Peterson [10]. . . . . . . . . . . . . . . . . . . . . . . 174 Illustration of the ported shroud. . . . . . . . . . . . . . . . . . . 185 Laminar and turbulent boundary layer . . . . . . . . . . . . . . . 236 h-s schematic of the ideal and real compression process. . . . . . 257 Velocity triangles on compressor wheel. Adapted from Sundstrom

[13] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278 Illustrative schematic of an idealized compressor wheel. . . . . . 289 An illustrative image of how a compressor map can look. . . . . . 3110 Default tab of mesh. . . . . . . . . . . . . . . . . . . . . . . . . . 3411 Sizing tab of mesh. . . . . . . . . . . . . . . . . . . . . . . . . . . 3512 Inflation tab of mesh. . . . . . . . . . . . . . . . . . . . . . . . . 3513 RSM residuals for the standard mesh simulation. . . . . . . . . . 3914 Illustration of the 3 design points included together with the de-

sign point taken from experimental data. . . . . . . . . . . . . . . 4315 Input and output parameters defined for use by the parametric

pack license. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4416 Location of inlet and outlet interface. . . . . . . . . . . . . . . . . 4517 Inlet-to-wheel domain interface. . . . . . . . . . . . . . . . . . . . 4618 Interfaces connecting wheel domain with adjacent domains. . . . 4619 Diffuser-to-wheel interface. . . . . . . . . . . . . . . . . . . . . . . 4720 Adiabatic geometry consisting of 3 domains, inlet, wheel and dif-

fuser /volute domain. . . . . . . . . . . . . . . . . . . . . . . . . 5021 CHT geometry consisting of 8 domains. . . . . . . . . . . . . . . 5022 Geometrical differences between CHT and adiabatic model. . . . 5123 Separately meshes of fluid and solid domains. . . . . . . . . . . . 5224 Final mesh of the solids and fluids of the CHT geometry. . . . . 5225 Pressure ratio and efficiency plotted as a function of inflation layers. 5426 Mass flow at outlet as a function of inflation layers. . . . . . . . . 5427 Mesh 3 in cross section. . . . . . . . . . . . . . . . . . . . . . . . 5628 The average and standard deviation of critical parameters . . . 5629 Efficiency compressor maps. . . . . . . . . . . . . . . . . . . . . . 5730 Pressure ratio compressor maps. . . . . . . . . . . . . . . . . . . 5831 Mass flow at wheel inlet interface . . . . . . . . . . . . . . . . . . 5932 Mass flow ratio, wheel inlet over compressor inlet. . . . . . . . . 5933 Mass flow ratio, ported shroud over compressor inlet. . . . . . . . 6034 Pressure recovery coefficient. . . . . . . . . . . . . . . . . . . . . 6035 Static pressure at ported shroud and 1 cm upstream in ported

shroud. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6136 Mean temperature ratio on measure lines on wheel inlet interface. 6237 Static temperature at wheel inlet interface. . . . . . . . . . . . . 6238 Total temperature at wheel inlet interface. . . . . . . . . . . . . . 63

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39 Cross section plane of temperature distribution in solids. . . . . . 6440 Two views to illustrate the temperature distribution on the com-

pressor wheel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6441 Comparison of temperature distribution, CHT model to the left,

adiabatic model to the right. . . . . . . . . . . . . . . . . . . . . 6542 Comparison of temperature distribution, CHT model to the left,

adiabatic model to the right. . . . . . . . . . . . . . . . . . . . . 6643 Efficiency comparison between CHT model, traditional adiabatic

model and experimental data. . . . . . . . . . . . . . . . . . . . . 6644 Pressure ratio comparison between CHT model, traditional adi-

abatic model and experimental data. . . . . . . . . . . . . . . . . 6745 Difference in power between the CHT and adiabatic model. . . . 6846 Total temperature comparison. . . . . . . . . . . . . . . . . . . . 6847 Total temperature comparison. . . . . . . . . . . . . . . . . . . . 6948 Static pressure comparison. . . . . . . . . . . . . . . . . . . . . . 6949 Static pressure comparison. . . . . . . . . . . . . . . . . . . . . . 7050 Total pressure comparison. . . . . . . . . . . . . . . . . . . . . . 7051 Total pressure comparison. . . . . . . . . . . . . . . . . . . . . . 71

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Nomenclature

Table 1: Parameters used throughout the report.

Variable Parameter Unit

Time t [s]Cartesian coordinates in i direction xi [m]Velocity component in xi direction ui [ms ]

Density ρ [ kgm3 ]

Stress tensor σ [ Nm2 ]External / body force F [N ]

Heat flux q [ Wm2 ]

Total energy e0 [ Jkg ]

Inner energy e [ Jkg ]

Specific heat at constant pressure Cp [ JK ]

Specific heat at constant volume Cv [ JK ]Enthalpy h [J ]

Entropy s [ JK·kg ]

Temperature T [K]Dynamic viscosity µ [Pa · s]

Kinematic viscosity ν [m2

s ]Angular velocity ω [rev/min]Reynolds number Re Unit less

Mass flow m [kgs ]

Tangential blade velocity ~U [ms ]Work W [J ]

Efficiency η [%]Pressure p [Pa]

Table 2: Sub and superscripts.

Subscript

Inlet of compressor 1Outlet of compressor 2

Reference value refSuperscript

Total condition 0Average

Favre average ˜Fluctuation ′

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1 Introduction

1.1 Background

Scania works continuously to develop internal combustion engines which canachieve low pollutant emissions and high efficiency. This requires designingcomponents with high performance and low weight while maintaining designconstraints and cost. Of high importance is also driver ergonomics, for examplenoise, and driving “feel”.

The capability to model compressor performance and characteristics is im-portant in order to have a virtual gas exchange design loop. Having this loopin place enable Scania to save time and/or cut development cost by excludingsome hardware acquisition from the design loop. Presently Scania does not havein place a fully robust simulation method for the compressor.

The background to this thesis is based on Scania’s demands to perform per-formance simulations that generates results with high accuracy and precision.Firstly to perform performance predictions that the department GT Power canuse for their engine performance simulations. Secondly to supply the depart-ment TMF with data and boundary conditions for their calculations of thermo-mechanical fatigue.

To meet this demands, Scania needs to improve their understanding of howthe performance of the turbo affects the engine. In the end, it is all aboutbeing an ”educated customer”, so Scania as the intermediate link can set highand righteous demands on their supplier of turbos. Therefore Scania needs tocontinuously improve their CFD methods.

1.2 Problem formulation

The problem formulations in general is to:

• Perform a mesh study by refining the boundary layer

• Compressor map surge line prediction using steady state simulations

• Conjugate heat transfer modeling

The first point is performed with the goal to refine the boundary layers inthe mesh to investigate parameters, and find a converging behaviour of theseparameters. This point is also performed to in terms of trade-off determine howa refined mesh will affect the computational time. Earlier work in this subjecthas been performed at Scania, but regarding the refinement of the volume. Thismesh study is focusing on resolving the boundary layers.

The surge line prediction is performed to get a better understanding of thephenomena surge. To investigate if there are any parameters that can indicate

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on the phenomena surge, by means of steady state simulations. Surge line pre-diction has been performed before at Scania using transient simulations. Thisinvestigation is performed purely with steady state simulations.

Conjugate heat transfer modeling is a modeling technique where the solidsare included in the geometry. The aim of this type of modeling is to investigatethe methodology of computing the heat transfer between the fluid and solid,but also between the solid and ambient using Ansys CFX. This has never beenperformed by Scania before, so the goal is both to investigate if it is achievable,and if so, will the aerodynamic performance of this method generate data closeenough to Scania’s traditional adiabatic method.

This thesis project is performed at Scania NMGD in Sodertalje. Because ofthe confidentiality in the work, all numerical plots will be normalized. This alsomeans that all graphical vector field is presented without legends.

1.3 Software

Ansys Workbench 18.1 is the software used throughout this thesis project, withthe CFD code CFX. In Figs. 1a and 1b below, the module used for CFDsimulations is shown together with the work flow.

(a) CFX module in Ansys Workbench 18.1 (b) Workflow in Ansys.

Fig. 1: CFX module and workflow in Ansys.

A geometry can be created by design modeler or spaceclaim. It can alsobe imported from an external CAD software used by i.e a designer. Whenthe geometry is done, it is dicretized using the mesh cell. In this step, a gridis put on the geometry to divide it into smaller cells, where every cell laterwill be solved for. After that, boundary conditions like pressure, temperature,rotational speeds, turbulence models etc, is included in the setup cell. Whenthis is done, the problem can be solved for in the CFX-Solver. The final step is

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to post process the result file from the solver. This is done in CFD-Post whereeither graphical vector fields for parameters can be used, or the data can beexported for numerical analysis in i.e Matlab.

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2 Theory

2.1 General Theory on centrifugal compressors

Because of ever increasing demands to increase engine efficiency and decreasevehicle emissions, the heavy-duty truck engine manufacturers are continuouslyinvestigating ways to satisfy both customers and legislation using old and newtechnology. Turbocharging is certainly not a new technology but is essential inorder to achieve high engine efficiency and the engine/turbo interaction is farfrom fully understood making turbocharging a highly active research topic forthe heavy-duty manufacturers.

Turbochargers works by letting waste gas from the engine be lead into theturbine housing. The gas will enter radially and start a rotation of the turbinewheel. The gas then exit the turbine axially and is lead away to the exhaustsystem. The turbine wheel is connected to the compressor wheel via a shaft,letting the energy from the rotating turbine wheel be transferred to the com-pressor wheel. The rotation of the compressor wheel will help to suck ambientair into the compressor axially. The air entering the compressor will flow downto the compressor wheel, where the rotation of the wheel will compress the air,and radially send it towards the volute (outlet). The now high pressurized gashas a increased density when passing through the air cooler and back into theengine. Modern direct-injection diesel engine combustion concepts are reliant onhigh air/fuel ratios in order to achieve high combustion efficiency. By using theotherwise wasted exhaust energy to achieve this, higher total engine efficiencycan be realized for the turbocharging.

The anatomy of the turbocharger is presented in Fig. 2 below.

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Fig. 2: The anatomy of a turbo.

2.1.1 Limitations

The domains of the turbocharger that this study will cover is the adjacent partsof the centrifugal compressor, that includes the inlet, impeller wheel, diffuserand volute. See Fig. 3 below for a more detailed view of the considered parts ofthe centrifugal compressor. This means that the turbine and bearing housing isnot considered in this study.

A centrifugal compressor consists of several parts. The inlet, where the airis passing through to the impeller (compressor wheel). The impeller consist ofseveral blades that will rotate around the axis and convert- the axial velocitiesof the fluid from the inlet into radial velocity. The inlet of the impeller is calledinducer, and the outlet of the impeller is the exducer. The air will be com-pressed in the wheel and with a radial velocity flow out of the impeller wheelthrough the diffuser. From the diffuser, the flow is then finally passing throughthe volute and outlet of the compressor.

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Fig. 3: Illustrative schematic of the centrifugal compressor. Adapted from Hill& Peterson [10].

There are different types of blades in the impeller wheel with different ad-vantages and usages. The type of impeller wheel studied in this master thesisconsists of 12 inclined blades. Six full blades and six splitter blades, alignedalternated around the circumference of the wheel. The full blades spans fromthe top of the wheel down to the diffuser, whilst the splitter is shorter in heightand starts further down the shaft. The combination of full and splitter bladesis a design that allows for higher airflow when operating at higher rotationalspeeds of the wheel.

The centrifugal compressor is limited in operating range by the surge andchoke lines respectively. The surge appears in the lower mass flow-rates of theoperating range whilst the choke appears at high.

2.1.2 Ported shroud

There are several methods to widen the operational range of a compressor. Twomain approaches are active and passive flow control. Active flow control includesvariable varying vanes installed in the diffuser and inlet guided vanes [8]. Bothof these methods widens the operational range of the compressor but has thedisadvantages of not being easy to design and produce. Passive flow control onthe other hand consists of methods such as different kind of casing treatmentslike the ported shroud.

The ported shroud is a cavity in the housing that allows the gas to recirculateback upstream. It is used to widen the operational range of the compressor, byletting gas pass through it at low mass flow rates. The reason for needing aported shroud is because of the phenomena surge that occurs in the vicinity ofthe compressor wheel. Surge or stall has the affect of letting the gas disrupt fromits natural flow, hindering the gas from passing through down to the compressorwheel. The cavity of the ported shroud can then be used to redirect the gasback upstream into the main flow and later flow down through the compressorwheel when the mass flow rate has increased. The widening of the compressor

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map will allow for the compressor to work at a greater span of mass flow rates.In Fig. 4 below, the ported shroud can bee seen.

Fig. 4: Illustration of the ported shroud.

2.1.3 Governing equations

Three essential equations are solved for in Ansys CFX in order to capture theflow characteristics and its evolution through the compressor. They are calledthe governing equations in the subject of fluid mechanics and are the unsteadyNavier-Stokes equations. The Navier-Stokes equations are the conservation ofmass (continuity), the conservation of momentum, and the conservation of en-ergy, and are defined below in cartesian tensor notation.

The continuity equation

∂ρ

∂t+

∂

∂xi(ρui) = 0 (1)

The momentum equation

∂ρui∂t

+∂(ρuiuj)

∂xj= − ∂p

∂xj+∂σij∂xj

+ ρFi (2)

The conservation of energy

∂(ρe0)

∂t+∂(ρuie0)

∂xi= −∂(pui)

∂xi− ∂qi∂xi

+∂(ujσij)

∂xi. (3)

Here, t = time, ρ = density, xi = cartesian coordinate, where i = 1, 2, 3,ui = fluid velocity component in xi direction, p = pressure, σ = stress tensor,

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Fi = body and external force in the system and qi = heat flux. In eq. (3),e0 = e + 1

2uiuj and denotes the total energy per unit mass, where e is theinner energy. An assumption is made that there is no external forces acting onthe system leading to Fi = 0.

The gas investigated throughout the thesis is air where the gas in assumedto be calorically perfect, so the specific heat at constant volume (Cv) and atconstant pressure (Cp) is constant. This gives the two relations:

e = CvT (4)

andh = CpT. (5)

Air is a newtonian fluid meaning that the viscous stresses emerging from thefluid is linearly. This is true at every point in the fluid, so a constitutive relationfor the newtonian fluid can be described as:

σij = 2µSij −2

3µSkkδij . (6)

In eq. (6) above, µ denotes the dynamic viscosity of the fluid, δij is theKronecker delta. δij is always in unity for i = j and zero for i 6= j. The rate ofdeformation, Sij is described as:

Sij =1

2(∂ui∂xj +

∂uj∂xi

). (7)

The gas will because of the compression in the system get a rise in temper-ature, which leads to the assumption that the dynamic viscosity is temperaturedependent. This is described by Sutherland’s law:

µ

µ0= (

T

T0)3/2(

T0 + TST + Ts

), (8)

where µ0 = 1.7231e − 5 [ kgm·s ] is the reference viscosity at T0 = 0◦C, andTS is the Sutherland’s constant.

2.1.4 Reynolds-Average Navier-Stokes

The above equations are used for simpler cases with steady laminar flow. How-ever, for the turbo compressor the flow is highly turbulent and chaotic. Thegeneral flow type throughout the compressor is in fact turbulent, with somesmall scale regions with laminar flow. In a turbulent flow, the flow parametersvaries with time in form of unsteady fluctuations. To capture the characteristicsof a turbulent flow, a method called Reynold’s decomposition is used, where eachquantity of the flow (i.e velocity or pressure) is split up into its time-averagedand time-dependent fluctuation part. For instance, the velocity:

u = u+ u′, (9)

19

where u denotes the time-averaged and u′, the fluctuation part. In additionto Reynold’s decomposition, when dealing with a compressible flow as in a com-pressor, density variations due to turbulent fluctuations has to be considered.This introduces the Favre averaging, which is a density weighted average thatsplits the previously mentioned quantity into a once again time-averaged andtime-dependent fluctuating part:

u = u+ u′′, (10)

where the difference between Reynold’s decompositon and Favre averaging isthe density weighted averaging applied in the latter, so the velocity u is relatedto the density via u = ρu

ρ .

Applying Favre averaging to the Navier-Stokes equations (eqs. 1-3) yieldsthe Reynold-Average Navier-Stokes equations (RANS equations):

∂ρ

∂t+∂(ρui)

∂xi= 0, (11)

∂(ρui)

∂t+∂(ρuiuj)

∂xj= − ∂p

∂xi+∂σij∂xj

+∂τij∂xj

, (12)

∂

∂t

(ρ(e+

uiuj2

) +ρu′′i u

′′j

2

)+

∂

∂xj

(ρuj(e+

uiui2

+ ujρu′′i u

′′i

2))

=∂

∂xj(ui(σij − ρu′′i u′′i )) +

∂

∂xj

(− qj − ρu′′j h′′ + σiju′′i − ρu′′j

u′′i u′′j

2

),

(13)

where τij = −ρu′′i u′′j is the Reynolds stress term, which is introduced in orderto capture the velocity fluctuations in all directions at macro scale. The Reynoldstress tensor is in fact six different equations leading to a closure problem sincethe system of equations to solve consists of more variables then equations.

2.1.5 Eddy-Viscosity modeling

To close this system of equations, the six components of the Reynolds stresstensor should be expressed in the averaging way described above. This typeof modeling, known as Eddy-viscosity modeling, will not introduce any newvariables and is a concept introduced by Joseph Valentin Boussinesq where theReynolds stress tensor and time-averaged velocity gradients are related to eachother as seen in eq. (14) below:

− ρu′′i u′′j ≈ 2ρνTSij −2

3ρνTSkkδij (14)

Here, νT is the turbulent eddy viscosity and represents the local propertyof the fluid. The Eddy-viscosity modeling is based on assumptions that limitsthe usage of it for turbulent flows. Firstly, the Reynols stress tensor can be

20

acquired from single point quantities and secondly, the strain rate tensor, Sijassumes that the behaviour of the time-averaged velocity gradients will have alinear dependency. This implies that the strain rate tensor will not be affectedby rotational velocities. Therefore it will be limited in centrifugal compressorswith swirling flows or boundary layer separations and is most suited for wall-bounded flows.

2.1.6 SST Turbulence model

Because of the limitations of the eddy viscosity modeling, flow models havebeen developed to better suit the turbulent flows found in turbo compressors.One of those are the Shear Stress Transport model that from now on will beaddressed as the SST model. The SST is based on the K − ω model and is thestandard turbulence used for simulations on centrifugal compressors because ofits ability to predict the flow separation under unfavourable pressure gradientthat are found in compressors. The model takes into account for the transportof the turbulent shear stress and take advantage of both the Wilcox and K − εmodel. It solves both transport equation for kinetic energy as well as specificdissipation rate. This makes the SST model more sophisticated and accuratethan its predecessors and is suitable for a larger range of different flows, likecompressors and aerodynamics.

The SST model is identical to the K − ε model in the free shear flows.At near walls, the prediction of separation from a smooth surface with highaccuracy is difficult, especially under unfavourable pressure gradients like in thecompressor.

It is recommended in the user guide to properly resolve the boundary layerwith at least 10 layers for the SST model to capture the flow correctly and withhigh accuracy at boundaries. This is especially of importance when solving forheat transfer [3]. Throughout this study, the SST model is used for simulationstogether with an automatic wall function.

This automatic wall function will switch from wall function to a low-Re nearwall treatment when the mesh is refined. As already mentioned, SST is recom-mended for accurate predictions near wall, but is mathematically identical tothe K−ε model in the free stream. By using the automatic near-wall treatment,the SST model will shift from a low-Re number form to a wall function formu-lation when the mesh is refined. This shift will be carried out to smoothly shiftbetween the two formulations, so the low-Re formulation is used for y+ < 2.This is why at least 10 layers of inflation should be used, so the wall can beresolved enough. Since the condition of y+ ≤ 2 is difficult to adapt in all partsof the geometry, the SST model with automatic near-wall treatment will shiftbetween low-Re and wall function for y+ < 11.06.

By resolving the mesh by the use of inflation layer, so the mesh is below thiscriteria, the simulation does not have to switch between different wall functions,which may lead to an increased accuracy for the solver.

21

2.1.7 Boundary layer

The Reynolds number in a centrifugal compressor is usually determined as:

Re =ωL2

ν, (15)

where ω denotes the rotational velocity and L is the height of the compressorwheel outlet diameter. For this compressor, the diameter is L = 99.5 mm. Ifmeasured at a rotational speed of ω = 109400 revolutions per minute (RPM ),and a temperature around 25◦C − 200◦C, the Reynolds number will be in theregions of 106 ≤ Re ≤ 107.

The high Reynolds number is mainly due to a low dynamic viscosity of thefluid (13.4 · 10−6 ≤ µ ≤ 35 · 10−6 m2/s). One might think that the low viscosityshould result in a negligible shear stress of the fluid. Because of the no-slipboundary condition at the walls, the viscosity is still of great importance.

Depending on the Reynolds number the flow in the boundary layer canbe in three stages. Laminar, transitional or turbulent. In the laminar case,the velocity profile is increasing linearly from the wall out to the boundarylayer edge and will have a more parabolic form than for a turbulent flow wherethe velocity profile is flatter. The laminar velocity profile most often have itsmaximum velocity in the middle of the channel with a magnitude around twiceof the average. For a turbulent flow, the velocity profile will instead have alogarithmic behavior, see Fig. 5 below. The slope will be more steep for theturbulent velocity profile which will lead to an increase in magnitude of the shearstress. The only exchange of mass and momentum of laminar boundary layerswill take place between adjacent layers. This exchange is not possible to witnesssince it occur on a microscopic scale. This exchange of mass and momentum willfor a turbulent boundary layer also occur between multiple layers and not onlyadjacent ones. Because of that, the exchange or mixing is possible to witnesssince it occurs on a greater (macroscopic) scale. In the boundary layer regionclosest to the wall (viscous sub-layer) the flow will however remain laminar evenfor a turbulent flow.

22

Fig. 5: Laminar and turbulent boundary layer

2.1.8 Performance

In order to quantify how well or efficient a centrifugal compressor is performinga set of parameters are introduced. The parameters that often is considered inthe subject is related to the air flow, pressure ratio and efficiency. Concerningthe air flow, parameters such as pressure (p), temperature (T ) and mass flow(m) changes are considered.

Since the geometry between two compressors can vary a lot, the best wayto express the performance of a compressor would be to use a dimensionlessparameter. This is where the pressure ratio is introduced. It is often describedin two terms namely total-to-total and total-to-static pressure ratio, see eqs.(16) and (17) below. The reason to use two ways to express the pressure ratiois mainly that the static way will use the stagnation pressure at the outletof the domain measured over, whilst total uses total pressure at the outletinstead. Total-to-total is the most common way to express the ratio but whenperforming test on the pressure, it can often be difficult to measure the totalpressure. Therefore both terms are used to express the ratio.

PRtt =p0

2

p01

, (16)

PRts =p2

p01

. (17)

The compressor wheel is applying energy to the fluid by the rotation of it.Sucking air into the inlet and compressing the air into the diffuser. The com-pressor is considered as a open system since air is transported into and outof the boundaries of the inlet and the volute (outlet). This allows for matter

23

in motion to be carried in and out of the system. The matter carries internalenergy with it as well as small scales of potential energy.

The law of conservation of energy states that the energy for an open systemmust be in balance:

∆Usystem + ∆Usurrounding = 0, (18)

thus with the first law of thermodynamics, the energy balance can be writtenas:

q +W = m∆(e+ pv +C2

2), (19)

where q and W on the left hand side is the heat transfer into and work doneon the system. The work done on the system is due to the compressor wheelrotating and is expressed in angular velocity multiplied with the torque fromthe shaft, ~ω · ~r. The heat transfer q is due to the energy from heat transfferedthrough the systems. Since the compressor wheel will produce much larger tem-peratures at the outlet than the solid domains, the heat transfer will work asheating on the solid domains. The ambient temperature of the outside walls ofthe solids are set to To = 45◦C, together with an wall heat transfer coefficientof htc = 35 [ w

m2K ]. This will contribute with a cooling effect by means of heattransfer energy on the solid.

On the right hand side of eq. (19) the changes in internal energy (e), work

required to move the fluid (pv) and the kinetic energy (C2

2 ) is multiplied withthe mass flow (m) to balance the equation.

e+pv will together form the specific enthalpy, h that with the kinetic energyadded expresses the total enthalpy, h0.

A common way to express the energy of the compressor is hence the equationbelow:

W = m(h02 − h0

1) = m∆h0. (20)

The compressor is assumed to be in a reversible process under specific con-ditions, so the total amount of heat added to the system can be expressed as:

ds =dq

T(21)

where s is the specific entropy. This lead up to:

dh = Tds− pdV, (22)

This is true also for a non-reversible flow since both the energy, entropyand volume are thermodynamic functions of state. For a compressor that both

24

are adiabatic and reversible, the specific entropy cannot change, and are calledisentropic. The efficiency for an isentropic compressor is expressed as:

ηis =Ws

W, (23)

where Ws denotes the work done for a constant entropy, and W is the actualwork into the compressor, W = h2 − h1. This is illustraded in Fig. 6 below ina schematic of the compression process.

Fig. 6: h-s schematic of the ideal and real compression process.

The schematic shows both the ideal and the isobaric (real) process. Theideal process is for a constant entropy and the isobaric process is determinedfrom the change in entropy of the fluid:

ds =dh

T− Rdp

p. (24)

In an isobaric process, the pressure remains constant, meaning dp = 0, so byintegrating eq. (24) yields the total enthalpy as:

h0 = h0ref · e∆s/Cp , (25)

Which defines the enthalpy as a function of entropy change. Since the effi-ciency as stated in eq. (23) is the the ideal work as a function of actual workthat defines the efficiency of the centrifugal compressor, the equation can berewritteen as:

25

ηis =Ws

W=h0

2,s − h1

h2 − h1. (26)

The parameter γ is introduced as the fraction of the specific heat at constanttemperature over specific heat at constant volume:

γ =cpcv. (27)

With the use of γ, the isentropic temperature ratio can be expressed as:

T2,s

T1= (

p2

p1)γ−1γ , (28)

which yields another expression for isentropic efficiency as:

ηis =(p2p1 )

γ−1γ − 1

(T2

T1)− 1

. (29)

As well as for the pressure ratio, also the efficiency is measured in term oftotal-to-total and total-to-static:

ηtt =PR

γ−1γ

tt

(T 02

T 02

)− 1, (30)

ηts =PR

γ−1γ

ts

(T 02

T 02

)− 1. (31)

2.1.9 Aerodynamics

The flow in the compressor is highly unsteady because of the presence of bothstationary and rotational parts, which makes it hard to investigate. A wayaround this problem is to use a reference system for the rotational part (com-pressor wheel), and by that have the ability to define the velocity triangles ofthe flow around the compressor wheel. The use of velocity triangles is a com-mon method for turbo machinery to represent velocity components of the fluidand can express both the absolute and relative velocity in the vicinity of thecompressor wheel.

Therefore a stationary reference frame is applied with its z-axis that coin-cide with the axis of rotation of the compressor wheel. In addition to that, arotating reference frame is also introduced with the same axis, meaning it willrotate around the z-axis with the angular velocity of the compressor wheel. Thevelocity triangle are expressed in cylindrical coordinates, so the power (W ) fromthe wheel generated onto the fluid can be formulated. It also has the benefit ofexpressing the radial and tangential velocity vectors.

In Fig. 7 below, the velocity triangles are illustrated both on the exducerand the inducer.

26

Fig. 7: Velocity triangles on compressor wheel. Adapted from Sundstrom [13]

Here ~U denotes the tangential blade velocity and represents the local velocityof the compressor wheel at a specific position ~r. Having the angular velocity ~ωof the wheel, ~U can be calculated as:

~U = ~ω × ~r. (32)

The absolute velocity ~c expresses the velocity in the stationary frame ofreference and ~ω is the relative wheel velocity of the rotating frame of reference.The work input per unit mass flow generated on the fluid from the compressorwheel can be determined by Euler’s equation for turbomachinery:

W = ω(r2Uθ2 − r1Uθ1), (33)

where r1 and r2 represents the radius of the inducer and exducer, as seen inFig. 8 below.

27

Fig. 8: Illustrative schematic of an idealized compressor wheel.

The flow enters axially at r1 and leaves radially at r2 for a centrifugal com-pressor. Eq. (33) for turbomachinery can also be written in terms of the localblade speed:

W1,2 = U2cθ,2 − U1cθ,1. (34)

By combining eq. (20) with eq. (34), the change in enthalpy can be expressedby the equation for conservation of rothalpy:

∆h0 =

∫ p2

p1

dp

ρ+

∫ T2

T1

Tds =1

2(U2

2 − U21 )− 1

2(ω2

2 − ω21). (35)

As seen in the above equation, the pressure rise is dependent for a changein enthalpy of the system.

2.2 Mesh dependency study

Besides turbo knowledge, a big part of CFD simulation demands high precisionin meshing to avoid errors. For that reason, a mesh study is great importance.The use of a mesh study is a way to prove the reliability of the mesh. If one wantsto publish a work at for example the organization ASME (American Society ofMechanical Engineering), the reliability has to be proven. A common standardto do this, is with the Richardson extrapolation. Therefore, as a part of themesh study, literature regarding Richardson extrapolation has been studied,where focus is on grid convergence criteria (GCI ).

The process of the mesh study is to generate different meshes, where everymesh will be more refined than the latter. Then by examining parameters ofinterest, one can study how much the function value of that parameter haschanged for the refined mesh. By using Richardson extrapolation, the goal isto examine the approximate relative error and the numerical uncertainty forthat parameter to numerically see how much the parameter has changed in

28

function value between the meshes. By examining the parameter graphically, itis possible to also illustrate the converging behaviour that the mesh refinementwill have on the parameter.

2.3 Surge line prediction

Surge is a phenomenon that has been around for as long as the turbo compres-sors. To this day, it is still a challenge to predict when it occurs with goodaccuracy. It is a widely studied problem, and regarding centrifugal compressorsin particular, work by Cumpsty [5], Semlithsch et al [8] and Nystrom [13] tomention a few, could be considered to better understand the phenomena.

Surge occurs at low mass flows, where the flow is connected to large systemoscillations or flow instabilities governed by high pressure gradients in the com-pressor. Fatigue from alternating stresses, both on the compressor blades andupstream components can occur during surge and this will limit the operationalrange of the compressor. In order to increase the operational range withoutrisk of going into surge, the compressor as well as upstream and downstreamcomponents must be designed with surge in mind. To achieve this, one has tounderstand why and when surge occurs. By increasing the knowledge of thephenomena, it is easier to avoid it to increase the lifetime of the compressor.

Surge is appearing at lower mass flow rates of the compressor, and is oftendetected by studying the mass flow. The best evidence of surge is an audibleturbulence [5]. When lowering the mass flow rate in the compressor, it is com-mon to get an increase in pressure rise. At some point, the pressure rise willreach its maximum value. For mass flows below this point, the compressor willgo into either surge or stall.

At this phenomena, the time dependent averaged mass flow will vary, com-pared to stall. This means that the compressor is changing between unstalledto stalled flow over time. This kind of behaviour is known as surge. The latterdescribed process can impact the compressor in such a powerful way that themass flow will indeed partially flow backwards leading already compressed gasto flow back out of the inlet. This process is known as deep surge. Deep surgemay however be acting in a more mild way where the operating point is nearthe top of the pressure rise. The time frame of surge (surge cycle) when thecompressor is acting in the regions of lower flow rates the, flow may temporarilybe stalling. At deep surge the oscillations are characterized by frequencies lowerthan the natural frequencies of the system.

The other type of surge is called mild surge, and is characterized by massand pressure oscillations governed by the Helmholtz resonance frequency. Atmild surge, reversed flow does not occur.

In general, surge is close to, or axisymmetric when fully developed. Initiallythe surge is not axisymmetric and this may have a huge impact on axial com-pressors where large transverse loads acts on the rotor, leading to wear on theblades.

29

A centrifugal compressor can still perform in presence of rotating stall. Bothaxisymmetric stall in the regions of the inducer tip and stationary non axisym-metric stall produced by asymmetry in the volute are operable conditions. Thereason for that is the fact that the compressor will still perform a centrifugaleffect on the gas leading to pressure rise even though it operates in stall.

When it comes to predicting surge, there are several uncertainties to con-sider. Uncertainties in form of geometry of the system and also the operatingconditions of the system.Regarding the geometry, studies have found that a sys-tem with larger volume downstream from the compressor often more easily canregister surge behaviour, in contrast to a smaller downstream volume [5]. Thedesign of the impeller blades such as tip and axial clearance, and also damage onthe blades in form of e.g. blade erosion has to be considered. For the operatingconditions, throttle changes in form of acceleration of the gas in the compressoris a contribution to uncertainty. Throttle change is partly due to the fact thata compressor seldom operates at a constant rotational speed, since the wastegas entering the turbine is varying in mass flow rate. Throttling is also affectedby gear shifting, due to an abrupt stop of mass flow entering the turbine duringgear shift. The uncertainties is the reason to introduce a concept called surgeline. This line is the margin that displays at what regions of the compressormap that surge occurs. Below the surge line, the compressor cannot operate ina stable regime. There are several definitions of where the surge line is situatedand fabricators of compressor often have their own definition. Because of theuncertainty in different surge line definitions, many producers use a save marginin the compressor map, often called surge control line. This is a great reasonfor getting a better understanding of surge and when it occurs, so the range ofthe compressor can be widened.

When setting up a compressor map, a parameter of interest (typically pres-sure ratio or efficiency) is plotted as a function of the mass flow rate that createsa speed line for a specific rotational sped of the compressor. This is done fordifferent rotational speeds. The common denominator of the different speedlines of the compressor map is that they all have a surge margin, as well as achoke margin on the other side of the mass flow range. The design point forevery speed line, where the maximum pressure ratio of that speed line is situ-ated is often linked together with a design line to display where the compressoris acting most efficiently for different rotational speeds. The surge line can belowered to widen the range of the compressor. This is achievable by changingthe shape of the blades, reducing the tip clearence or perform changes of thecasing [5](s.367).

To predict the behaviour of surge, one needs to be reminded that a com-pressor can work satisfactory even though a vast part of the flow is separated.Because of the fact that a centrifugal compressor wheel generally have severalzones of stall or separated flow even though the compressor isn’t in a operationpoint of surge, there is a difference in opinion whether or not rotating stall is anorigination of surge. Also the fact that the components should be investigatedas a system and not treated as several separated components. According toCumpsty [5], the components of the compressor should be treated as a coupled

30

system, so the combined effect of the parts is to be considered in the aim ofpredicting surge.

Manufacturers of turbos often illustrate where the phenomena surge is bypresenting it in a compressor map. A compressor map is often shown as thepressure ratio or efficiency of the compressor as a function of mass flow at theoutlet. In Fig. 9 below, an example of how a compressor map can look, ispresented.

Fig. 9: An illustrative image of how a compressor map can look.

This compressor map is presented as pressure ratio as a function of massflow rate for different speed lines. Every speed line corresponds to a certainrotational velocity that the compressor is operating at for mass flows rangingfrom surge (low mass flow) to choke (high mass flow).

Five speed lines means that the compressor is tested for five different rota-tional speeds. The bold line on the left connecting the speed lines is the surgelimit line. It tells at which mass flow rate the compressor is possible to operateat the different speed lines. In the other end of the speed lines is the choke limitline. This phenomena will not be managed in this thesis project. The blackdashed line correspond to where the compressor works most efficient for all the

31

tested speed lines. Finally, the red surge control line, is a safety margin thatsome manufacturers includes in their compressor map to illustrate where thecompressor should be operated above, to avoid damage. The surge control lineis not used by all manufacturers, and there is no general equation of where itlies, so it rather is the manufacturers own choice of safety margin.

2.4 Conjugate Heat Transfer

The aim of this part is to investigate the methodology of setting up a ConjugateHeat Transfer (CHT ) model for CFD simulation, to solve heat transport be-tween fluids and solids, but also between solids and ambient. Also to investigatehow the CHT model aerodynamic performance will do in comparison againstthe traditional adiabatic CFD method, as well as experimental data.

When performing CFD simulations on centrifugal compressors, Scania usu-ally use the traditional method of only considering the fluid components of thegeometry and apply adiabatic walls as boundary condition. This method isfairly straight forward when it comes to solving.

The CHT model introduces conducting solids to the model that surroundsthe fluid domains. This setup visually gives a better understanding on how thedifferent domains is interacting with each other. The method is however morecomplex to implement because of the increased number of domains to consideras well as the number of interfaces and boundary conditions that has to beincluded in the setup. Between every domain (solid to solid, solid to fluid orfluid to fluid) an interface is needed (on both sides) to connect the domainsproperly. The amount of interfaces subjected to frame change is increased byusing CHT, so to effectively set up the case, it is essential to pre-define whichfaces that should be considered as a interface. This will speed up the processin CFX Pre since a named selection is easier to use than the auto generatedsurface names created by Ansys. The autogenerated names are typically calledFXXXX.XXXX, where X is some arbitrary number between 0 and 9. It is avital part of the modelling to define all this interfaces. If not done, Ansys CFXwill automatically apply a wall where an interface is not defined. Not only that,but also per default set it as adiabatic, which is not preferable when solving forheat transfer.

When performing a CHT case, CFX will inlcude terms for heat transport,conduction and volumetric heat sources into the energy equation. This is doneautomatically when domains are specified as solids in the geometry.

The benefit och using a CHT method is the ability to determine heat trans-fer characteristics of the simulation such as heat flux through between fluid andsolid parts, or the heat transfer coefficients of the material. Information of heattransfer characteristic is of great importance because of reduced engine sizesand increased exhaust gas temperatures. By performing a CHT analysis, heatflow distribution can be analyzed in both the fluid and the solid. Knowledge

32

of this parameters can be used by engineers when testing compressors in labo-ratories in order to asses more trustworthy data. The parameters can also beused in order to calculate stress and fatigue of the compressor in Ansys’s othersimulation environments, such as Ansys Mechanics.

There are also drawbacks with CHT methods. For instance, the geometryneeds to be created both for the fluid and solid parts, which is a time demandingprocess. The mesh will also gain an increase in generation time since both solidand fluid parts needs a mesh.

33

3 Method

3.1 Mesh Independency study

3.1.1 Geometry

The first problem considered, is to import the geometry and check for problemswith it. The geometry consisted of three parts, namely the inlet, wheel and dif-fuser. In Ansys, the usual tool for editing a model is by using Design modeler.For geometry checks, there is however more convenient to use the second tool forgeometries which is Spaceclaim. In Spaceclaim, the different parts of a modelcan be highlighted and checked for problems. For the compressor investigatedthere was a lot of problems concerning the boundaries. By checking which facesof the parts that had problems and cut out and paste back the faces, a builtin stitch feature was used to sew the geometry back. This method would takeaway all boundary problems that Spaceclaim flagged about. The same proce-dure were performed for all three parts of the geometry.

3.1.2 Meshing

When the geometry was checked and problem free, the mesh could be set up.At Scania, there are some standard settings when setting up the mesh, which ispresented below.

Fig. 10: Default tab of mesh.

As seen in Figure 10, the solver preference is set to CFX with relevance at66. The relevance is an general option to control the fineness of the mesh, andby setting it to a value, it will have that relevance through the entire model.Scania is using 66 as a default value in Ansys.

34

Fig. 11: Sizing tab of mesh.

The next tab is sizing, see Figure 11. Also for sizing, there are default settingsthat Scania uses, like size function, that is set to proximity and curvature. Thisoption will allow the mesher to minimum number of element layers in regionsthat constitute gaps. It also examines the curvature on edges and faces andcomputes element sizes on these entities such that the size will not violate themaximum size or curvature normal angle. There is also a Scania standard touse Min size = 5e− 4, proximitiy min size = 2.5e− 4, max face size = 2e− 3and max tet size = 3e− 3.

Fig. 12: Inflation tab of mesh.

In Figure 12 above, the inflation tab is presented. The inflation layer is usedto be able to have a great refinement near walls. This is a vital point in order tocapture the viscous effects that rise and has a significant roll of the flow, closeto no-slip walls. First layer thickness works by defining the first layer height,meaning what height the first inflation layer will have from the wall. Then byvarying the growth rate and number of layers, a total inflation layer height willbe produced.

A key feature of inflation is which method used for collision avoidance. Thisis once again a Scania default to use layer compression, which task is to com-press inflation layers in areas of collision. The defined heights and ratios in theseareas are reduced to guarantee the same number of layers throughout the entire

35

inflation region.

Having the basic mesh tabs set up for wanted options, the next task is toassign all included parts in the mesh with suitable settings to assure great meshdesign. There are three settings used with different refinement for all three parts.Patch conforming method, inflation and face sizing. Patch control method isa method control that provides support for 3D inflation with a built-in growthand smoothness control. This method cannot be interpreted for the whole ge-ometry, so it is inserted for all three parts of the model.

The inflation is used to assign which faces of a part that should have infla-tion layers. Often a whole part is considered, except the interfaces that connectsdifferent parts. It is especially of importance to use inflation at faces with a lotof curvature or where a higher refinement is required to capture the flow char-acteristics, i.e the blades of the wheel.

The face sizing is used to control the adjacent faces to the faces within thescope, which means that meshes on the adjacent faces will transition smoothlyto the size on the scoped face.

A key part of the meshing process is to meet the design criteria that Scaniahave to not have a higher skewness of the mesh than 0.95. The criteria of 0.95 isbased on experience at Scania as a best practice guideline and is not an officialdemand. Skewness is a quality measure that determines how close to ideal aface or cell is. It ranges from 0.0 to 1.0, where values above 0.9 is consideredto be bad. One has to take into account that the geometry of the turbo hassome extremely complex parts, such as the blades. For that reason the criteriaof values below 0.95 is a sacrifice that has to be made. A lower skewness isachievable but with the result of a high cost in computational time. For thatreason, a sacrifice has to be made. As long as the elements of high skewnessis not located at important locations of the domain that will consist of flowcharacteristics that needs to be captured, it is acceptable to keep some elementsof higher skewnwess.

One way of dealing with high skewness is by using a structured mesh insteadof the unstructured tetrahedron mesh. This method uses a block-structuredgrid that easier controls the mesh cell distribution and is highly space efficient.Since the geometry of the turbo is very complex and this mesh study focuseson the boundary layer and the outcome of refining it, an unstructured mesh isconsidered. Furthermore, Scania has through experience found out that CFDsimulations with unstructured mesh will produce just as good result as for astructured mesh.

To determine how refined your mesh needs to be and still be able to gen-erate expected result, a mesh study is performed. Similar work has been doneearlier [14] regarding a mesh study on the volume mesh. From that study,

36

the conclusion drawn was that a mesh of 13, 887, 787 number of cells was thebest choice in order to capture the behaviour of the fluid correctly and with aslow computational effort as possible. The aim of the mesh study is to find agrid convergence index (GCI) small enough to ensure that a more refined meshwould not change the solution anymore. The GCI tells in percentage the nu-merical uncertainty a certain parameter has, compared between two meshes [12].

As mentioned earlier, for this mesh study, the focus is on refining the bound-ary layer. This is done purely with the inflation option First layer thickness.

The results from a simulation will partly depend on the discretization of thedomain but also on the approximation of the mathematical model. Both trun-cation errors and round-off errors will contribute to the solution leading to adifference between numerical and exact solution. Truncation errors is the differ-ence between the discretized equation and the exact one and is often estimatedby replacing the nodal values in the discrete approximation by a Taylor seriesexpansion about a single point. The discretized equation will consist of the orig-inal differential equation and a remainder which corresponds to the truncationerror. For the discretization to be consistent, one has to avoid truncation errors,which is done by letting the mesh spacing approach zero.

The round-off error is a form of a quantization error that occurs due to therepresentation of floating point numbers on the computer and limits the accu-racy at which the the computer stores numbers.

The generated result from simulations needs to be validated to ensure goodquality with high precision when solving. This is where the mesh study is intro-duced, to validate if the numerical and physical approximations from the solveris good enough to capture the flow characteristics of the domain.

The first step in the process before analyzing the discretization error is toshow that the iterative convergence of the solver has preferably four orders ofmagnitude decrease in the normalized residuals. This needs to be fulfilled forevery equation solved for. ASME Journal of Fluids Engineering has a criteria[11] of at least an order of three, but to be able to capture the flow characteristicscorrectly, a better convergence criteria (smaller) can be justified if the time costis within reasonable frames. Usually, the convergence criteria is therefore set to1 · 10−4. A common method for discretization error estimation is Richardsonextrapolation (RE), which is also the method that ASME recommends. Themethod has been around since 1910 [7] and is a commonly used standard. Ithas its limitations but is the most reliable method available for numerical un-certainty prediction to this date.

For this study, mainly three different meshed will be investigated, but toeasier find the convergence to an asymptotic numerical value, two additionalmeshes are introduced. As mentioned earlier the volume mesh will not be ac-tively treated, since the goal of the study is to investigate how the boundary

37

layer refinement will affect the result. That means that different approachesof inflation layer usage will be the method in focus. For all five meshes, thepreviously mentioned inflation layer method First layer thickness is used. Theapproach was to keep the first layer height constant to 3·10−6 m, and by varyingthe growth rate and number of layers to try to achieve a fairly constant totalinflation layer height between the five meshes.

Throughout this study, the subscript 1, 2 3 4 & 5 is used declaring therefinement of the mesh, where a lower subscript indicates a more refined mesh.The used parameters are seen in Table 3. Mesh3 will because of its great usagein the thesis also be refered to as the standard mesh.

Table 3: Parameters used for the three meshes.

h1 [m] Number of layers GR [%] htot [m]Mesh5 3 · 10−6 5 2.9445 3.9994 · 10−4

Mesh4 3 · 10−6 9 1.743 3.3993 · 10−4

Mesh3 3 · 10−6 10 1.5 3.3999 · 10−4

Mesh2 3 · 10−6 16 1.2284 3.3994 · 10−4

Mesh1 3 · 10−6 18 1.188378 3.3993 · 10−4

where h1 denotes the first layer height, GR is the growth rate and htot isthe total inflation layer height. In Ansys CFX Manual, a recommendation whenusing inflation methods is to use at least 10 layers or more to properly resolvethe boundary layer, and capture the flow characteristics close to the wall. Forthat reason, it will be of greater interest to compare Mesh3-Mesh5.

Below, the residuals of the simulation is presented for the standard mesh of37, 915, 257 elements. The chosen design point for the simulation is a mass flowof m = 0.2 [kg/s], ω = 87000 [RPM ], that is run at the inlet temperatureTref = 298.15 [K] with relative pressure of pref = 100 kPa.

38

0 100 200 300 400 500 600 700 800

Accumulated time step

10-6

10-5

10-4

10-3

10-2

Variable

valu

e

RSM Residuals

P-Mass

U-Mom

V-Mom

W-Mom

Fig. 13: RSM residuals for the standard mesh simulation.

As seen in Fig. 13, the U-, V- and W-momentum residuals are from approx-imately 450 time steps oscillating around the iterative convergence criteria of1 · 10−4. Besides the iterative convergence criteria, a solution control is set toterminate the solution after 800 accumulated time steps. The more time stepsused, the longer the simulation will take, so this limit of 800 is also a value basedon experience at Scania from earlier works. By setting the value to a highernumber of maximum time steps, one might achieve residuals that converges tovalues below 1 · 10−4. However, for this mesh study, the solution is consideredto be converged enough. The fourth function in the figure corresponds to theP-mass residual and it is noticeable that it converges faster than the three otherswithout any problems.

The RE method is consisting of several steps. First of all one has do definea representative cell size. For a 3-D problem, the approach is done using eq (36)below,

h =[ 1

N

N∑i=1

(∆Vi)], (36)

where ∆Vi is the volume of the ith cell and N corresponds to the number ofcells in your mesh. In Ansys Meshing, there is an inbuilt quality tool that letsyou investigate the cell characteristic length for the mesh. This length is theone that is used as the representative cell size h, and it is presented for all themeshes in Table 4 below.

39

Table 4: Mesh statistics

Number of elements h [m]Mesh5 29, 579, 983 3.8392 · 10−4

Mesh5 34, 445, 237 3.2874 · 10−4

Mesh4 37, 915, 257 2.9869 · 10−4

Mesh2 48, 571, 877 2.3326 · 10−4

Mesh1 52, 111, 560 2.1747 · 10−4

The second step is to verify that the grid refinement factor r is greater than1.3 which is a recommended value generated from experience [11]. The gridrefinement factor is the ratio between the coarse and refined characteristic celllength. Therefore, the grid refinement factor is calculated using Mesh5 andMesh1 as the course and refined mesh respectively. The calculated refinementfactor is presented in eq (37) below.

r =hcoarsehrefined

=h5

h1=

3.8392 · 10−4

2.1747 · 10−4= 1.7654 (37)

The grid refinement factor is greater than 1.3, hence the RE method proceedsto step 3 where you first define the variables r21 and r32,

r21 =h2

h1= 1.3735, (38)

and

r32 =h3

h2= 1.2853. (39)

The apparent order p of the method is calculated using eq (40a-40c) below,

p =1

ln(r21)

∣∣∣ln∣∣ε32

ε21

∣∣+ q(p)∣∣∣ (40a)

q(p) = ln(rp21 − srp32 − s

)(40b)

s = 1 · sgn(ε32

ε21) (40c)

where ε32 = φ3 − φ2, ε21 = φ2 − φ1 and φk denotes the solution of thekth grid. φ is the variable of interest, which of this study consists of mass flow,pressure ratio and efficiency.

Initially, one has to guess the apparent order p by setting it to be the firstterm of eq (40a) and neglecting the second. Eq (40b-40a) is computed iterativewith the initial guess of p for the first iteration and the updating it for everystep until p gets stable.

40

Step 4 of the process consists of calculating the extrapolated values from

φ21ext =

rp21φ1 − φ2

rp21 − 1(41a)

φ32ext =

rp32φ2 − φ3

rp32 − 1. (41b)

The fift and final step is to calculate error estimates, along with the apparentorder p. This can be done both for approximate relative error,

e21a =

∣∣∣φ1 − φ2

φ1

∣∣∣, (42)

and extrapolated relative error,

e21ext =

∣∣∣φ21ext − φ1

φ21ext

∣∣∣. (43)

The extrapolated relative error is a discretization uncertainty between twomeshes (coarse and standard, or refined and standard mesh).

Finally, the fine-grid convergence index GCI21fine

GCI21fine =

∣∣∣1.25e21a

rp21 − 1

∣∣∣ (44)

The latter corresponds to a numerical uncertainty of the fine grid, for thedifferent variables of interest φ, between two meshes.

41

3.2 Surge line prediction

Experimental study has been performed on the SC92T70 compressor in PanklTurbosystems laboratory located in Mannheim Germany. The laboratory hasperformed test on several operation points. The Different rotational speeds usedvaries from approximately 41000 to 116000 RPM. Where different mass flowshave been used ranging from surge to choke line. For this thesis, 4 differentoperational speeds has been picked from the experimental dato to produce acompressor map with CFD simulations. The laboratory performing the experi-mental data had some deviation for the different examined rotational velocities,so the velocities chosen for CFD simulations are takes as the average. Theserotational velocities are 56250, 68000, 87000 and 109400.

The experimental data is ranging from surge to choke, but not in the regionof surge. This is due to the laboratory not having the ability to perform testin this region. Therefore three design points per speed line is included thatis beneath the region of surge in mass flow rate for this CFD analysis. Byincluding these design points with the ones from experimental data, it shouldfrom post processing of CFD data be able to find parameters that can indicateon the phenomena. These three design points are shown as an illustration inFig.14

42

Fig. 14: Illustration of the 3 design points included together with the designpoint taken from experimental data.

The model used for the map was the standard mesh of the mesh indepen-dency study (Mesh 3). When performing a compressor map, a initial run isperformed to retain data from a case known to converge. In Ansys post proces-sor (CFD Post), expressions is set up to monitor different flow characteristicsthrough the compressor system. These expressions are based on surfaces, linesand points of calculations.

An expression in CFD-Post can be checked as an input or output parameter.When performing the initial run, the exit-corrected mass flow was used as aninput parameter. The input parameter is then constraining the solver with thatmass flow as a boundary condition. For the compressor map, expressions ofinterest are declared and set as output parameters instead. The compressormap will then calculate all declared output parameters for every design point.

A compressor map is solved using Ansys HPC Parametric Pack licens, whichis a license option offered to be used for Ansys workbench jobs. The benefit ofthe license is to compute multiple design points in parallel for a single designstudy. In the license, a total of five HPC parametric pack can be utilized.The first parametric pack allows for four simultaneous jobs to be calculated inparallel, and for every addition of parametrick pack license, the previous amount

43

of simultaneous jobs to be calculated is doubled. A total of four parametricpack licenses has been used for this simulation, resulting in 32 simultaneousdesign points being calculated in parallel [4]. This means that for every designpoint calculated, all declared output parameters for that particular case will becalculated.

For solving the compressor map, the parametric pack is initialized by solvingfor the current design point, meaning the previously solved design point usedto set up the compressor map. This is due to the fact that from the results ofit, it can be ensured of convergence. Knowing that your simulations of multipledesign points is initialized with a design point known to converge is of greatimportance, since the solver uses parts of the already solved information whensolving for the later ones. Therefore, it is more efficient for the solver to solveall design points at a certain rotational velocity before solving for the next [1].In addition to that, for a certain rotational velocity, the order of the designpoints to solve should be set to start from a point known to be near maximumefficiency, since it will speed up the convergence.

By performing a compressor map, the settings used for the different designpoints will be the same except for the mass flow and rotational speed. Thismeans that the setup used only needs to be defined once, and then configurethe parametric pack to calculate for the varying mas flow and rotational veloc-ity. The following setup for the simulation is therefore the setup used for allcases, except for the mass flow and rotational velocity. In Fig. 15 below, a viewof the input, and output parameters can be seen.

Fig. 15: Input and output parameters defined for use by the parametric packlicense.

The two first columns that are circled, ecmflow and wheelspeed corres-nponds to the input parameters used for the parametrick pack. To the right ofwheelspeed the outputparameters are seen. These are defined in CFD-Post. Byvarying the input parametes ecmflow which is the exit corrected mass flow, a

44

whole speed line can be solved for. Exit corrected mass flow will be explainedlater in section 3.2.1. By varying wheelspeed which is the rotational speed,multiple speed lines can be solved for. For example, ten different mass flows,and four different rotational speeds will generate a total of 40 different inputs tothe solver. A total of 140 different output parameters are defined, so when allinput parameters are solved for, a big table will be generated with data of theoutput parameters. These output parameters are then later examined in orderto find indications of the surge phenomena.

As mentioned in section 5.1, the mesh chosen for later simulations arethe standard mesh (Mesh3) of 37, 915, 257 elements. The inlet and outlet isextruded 8 diameters to ensure fully developed ans stable flow in the system.Boundary condition for inlet and outlet are visualized in Fig. 16 below.

Fig. 16: Location of inlet and outlet interface.

An interface is connecting the inlet and wheel domains.

45

Fig. 17: Inlet-to-wheel domain interface.

For the wheel domain, three interfaces are used to link the fluid from wheelto inlet, ported shroud and diffuser respectively as illustrated in Fig. 18 below.

Fig. 18: Interfaces connecting wheel domain with adjacent domains.

The top red interface is wheel inlet interface, middle red is the ported shroudinterface, and the bottom red is the wheel outlet interface (also known as wheel-to-diffuser interface). These interfaces will later be of interest when investigatingthe surge phenomena.

When performing a steady state simulation including a moving referenceframe (MRF ) that is connected to a stationary with an interface, must be setup using the proper frame changing model. In Ansys CFX-Pre, three differentoptions are available as well as the option to disable frame change. Earlier workon compressors have proven that the best suitable frame change option for thiskind of problem is Frozen rotor [14]. This model works by letting the frame of

46

reference change, but let the relative orientation of all the components acrossthe interface be fixed. This is done by letting both the frames of reference con-nect so they keep a fixed relative position throughout the solving.

Disregarding the outlet interface of the volute domain, one interface is usedto connect the diffuser with the wheel domain. It is shown in Fig. 19 below.

Fig. 19: Diffuser-to-wheel interface.

The analysis type is steady state, meaning it will use Reynolds AveragedNavier Stokes equation, mentioned earlier in section 2.1.4. Air (compressible)is used as fluid with the thermodynamic state of gas. The gas will reach speedsover Mach number Ma = 0.3, and is therefore modelled with total energy. Thisenergy model differs from the usual Thermal energy just by taking into accountthat the velocities will reach subsonic and transonic velocities. Defined datafor air is presented in Table 5 below, where T is temperature, ρ is density, κis thermal conductivity and Cp is specific heat at constant pressure and µ isdynamic viscosity.

47

Table 5: Air data for different temperatures.

T, [C] ρ, [kg/m3] κ, [W/mK] Cp, [J/kgK] µ, [Ns/m2 · 10−6]-150 2.866 0.01171 983 8.636e-060 1.292 0.02364 1006 1.729e-55 1.268 0.02401 1006 1.754e-510 1.246 0.02439 1006 1.778e-515 1.225 0.02476 1007 1.802e-520 1.204 0.02514 1007 1.825e-525 1.184 0.02551 1007 1.849e-530 1.164 0.02588 1007 1.872e-535 1.146 0.02625 1007 1.895e-540 1.127 0.02662 1007 1.918e-545 1.110 0.02699 1007 1.941e-550 1.093 0.02735 1007 1.963e-560 1.060 0.02808 1007 2.008e-570 1.029 0.02881 1007 2.052e-580 1.000 0.02953 1008 2.096e-590 0.9720 0.03024 1008 2.139e-5100 0.947 0.03095 1009 2.181e-5120 0.898 0.03235 1011 2.264e-5140 0.8544 0.03374 1013 2.345e-5160 0.815 0.03511 1016 2.420e-5180 0.779 0.03646 1019 2.504e-5200 0.745 0.03779 1023 2.577e-5250 0.675 0.04104 1033 2.760e-5300 0.617 0.04418 1044 2.934e-5600 0.404 0.06093 1115 3.846e-52000 0.155 0.11113 1264 6.630e-5

3.2.1 Boundary Conditions

By using a boundary condition for the pressure or for the temperature at theoutlet to control the mass flow, it is hard cover the whole mass flow rate of aspeed line without entering surge or choke. Therefore it is common to use anexit corrected mass flow, that corrects the mass flow rate to total conditions atthe outlet. With this method, the exit corrected mass flow can be maintainedconstant. It relates the mass flow with pressure and temperature at the outlet aswell as the reference pressure and temperature at the inlet. Besides the benefitof not having to change the total conditions, it also makes it possible to beused for different compressors, since it is adjusted with reference pressure andreference temperature. Therefore two compressors operating at different massflows can use the same exit corrected mass flow. The equation for exit correctedmass flow is:

48

mec = m

√T 02

Tref

p02pref

. (45)

The outer walls of the fluid domains are modelled as adiabatic walls. Thisis a standard boundary condition that Scania uses for their CFD simulations ofcentrifugal compressors. By using adiabatic walls, no energy will pass throughthe walls, they are fully isolated. This means that the only energy contributionto the system is the energy that the rotating compressor wheel will transfer tothe fluid in form of heat and kinetic energy.

As mentioned above, a total of 140 output parameters where defined toexamine the phenomena to find indications of surge. The ones found will bepresented in section 4.2.

49

3.3 Conjugate Heat Transfer

For the mesh study and the surge line prediction, a geometry of the fluid partswere imported from the designers at Scania. This geometry will also be calledas the adiabatic or traditional geometry later on in this report. For the CHTcase, a geometry was once again imported from the designers, but this time forthe solids. Therefore the fluid parts had to be created using Spaceclaim. Thefluid geometry can be seen in Figs. 20a and 20b below.

(a) Adiabatic geometry viewed from theside.

(b) Cross section of the adiabatic geometry.

Fig. 20: Adiabatic geometry consisting of 3 domains, inlet, wheel and diffuser/volute domain.

In Fig. 21a and 21b below, the CHT geometry is presented.

(a) CHT geometry viewed from the side.(b) Cross section of the CHT geometry.

Fig. 21: CHT geometry consisting of 8 domains.

As seen in the Fig. 21b to the right, the CHT geometry consists of 8 domains.The fluids in pink, compressor house solids in green, compressor wheel (solid)in blue, and finally shaft and shaft nut in gray.

50

Fig. 22: Geometrical differences between CHT and adiabatic model.

Even though the geometry imported from the designers used for the two firstproblems in this thesis should nominally be the same as the geometry for theCHT problem, there was geometrical differences on several locations around thecompressor wheel. In Table 6 below, these differences are presented.

Table 6: Geometrical differences between CHT and adiabatic geometry

CHT geometry Adiabatic geometryTrailing edge, [mm] 0.4002 0.3999dr1r2 , [mm] 29.7874 30.1061dps, [mm] 1.9997 1.4344hps, [mm] 3.0113 5.8780hW2D, [mm] 4.3561 4.3267

Here, Trailing edge corresponds to the distance from the end of the blade(closest to the wheel outlet) to the shroud. dr1r2 is the distance from the shaftto the shroud. dps is the radial distance from the shroud to the ported shroudinterface. hps is the height of the ported shroud interface. hW2D correspondsto the height of the wheel-to-diffuser interface.

When it comes to meshing, the method was to divide the parts into twomodules. One for fluids and one for solids. The benefit with this approach isto have global mesh settings that will apply for all solids, and vice versa for thefluids. Ansys is good at writing big files of data for every CFD projects, so bymeshing solids and fluids separately, the memory usage in the computer can bereduced. In Fig. 23 below, the two mesh modules are illustrated. The imagealso shows how the two mesh modules later links together into the setup cell.

51

Fig. 23: Separately meshes of fluid and solid domains.

One drawback of the separate mesh modules is the interface generation. Thishas to be set up for all interfaces, connecting two faces, and needs to be doneon both sides to link them together for heat transfer. This is a time demand-ing process that in the end lead up to 74 interfaces. By having only one meshmodule, it is easier to keep track of all interfaces and ensure that both sides ofthe interface are defined.

For mesh settings, the fluid parts are meshed with the exact same settingsas for the mesh study and surge line case. The solids will however not havethe same refinement due to the linearity of heat transfer in solids. The solidsare therefore meshed with default settings, and without any refinement of theboundary layer. The final mesh of the solids and fluids is presented in Figs.below.

(a) Mesh of solid domains. (b) Mesh of fluid domains.

Fig. 24: Final mesh of the solids and fluids of the CHT geometry.

In the surge line case a frame change option was introduces to connect MRFfaces with stationary faces on the wheelfluid. In the CHT case, in additionto the wheel fluid, also the wheel solid, shaft and shaft nut is rotating, whichmeans that a lot more frame change interfaces must be defined. This is anotherbenefit of having pre defined all faces with named selection, to easier connect

52

them in CFX Pre.

All solids must also have the right thermodynamic data, such as thermalconductivity, specific heat and density in order for the solver to properly com-pute the heat transfer in the solids. The outer walls of the traditional case weremodeled with adiabatic walls. In this model the outer walls are modelled withan ambient temperature of Tout = 45[◦C] to correspond to the temperature inthe engine room. The outer walls is also set up with a heat transfer coefficientof htc = 35 [W/m2 ·K], to allow heat transport out of the solid walls.

Since the CHT model also includes the rotating shaft, that will rotate inclose vicinity to the compressor house, friction and thermal energy will arise inthis region. This is modelled by introducing a thin layer of air (0.5 mm) thatwill lie between the shaft and compressor house as a thermal resistance.

Another difference in the setup compared to the traditional adiabatic case isthe time step used for the solver. Normally when performing CFD simulationson a compressor, a physical time step is used of 1/ω, but for this CHT case, anautomatic time scale is introduced, with a timescale factor of 10 for the fluidparts, and a timescale factor 100 for the solid parts, [6].

Other than that, the CHT model is identical in setup compared to the tra-ditional adiabatic model. The simulations were performed on one speed line of109400 RPM . The solid mesh ended at 5, 754, 398 elements, and the fluid on38, 844, 114, leading to a total of 44, 598, 512 elements of the final mesh. Thesimulation time for one design point on this speed line took 7h 27min runningon 168 CPUs.

53

4 Results

4.1 Mesh independency study

In Fig. 25a, 25b and 26 below, a graphical view of the examined parametersare shown. Here, the parameters are plotted as a function of refined mesh, toillustrate the converging behaviour resulting of the mesh refinement.

(a) Total-to-total & total-to-static efficiency. (b) Total-to-total & total-to-static pressure ratio.

Fig. 25: Pressure ratio and efficiency plotted as a function of inflation layers.

Fig. 26: Mass flow at outlet as a function of inflation layers.

With the Richardson extrapolation method, the approximate relative errorand numerical uncertainty (GCIfine) is calculated for the above parameterscomparing Mesh3 and Mesh4, with the steps presented in Section 3.1.2. The

54

table also present the simulations time used for solving each case, using a specificammount of CPU cores. These are presented in table 7 below.

Table 7: Approximate relative error and numerical uncertainty for examinedparameters between Mesh3 & Mesh4.

m ηtt ηts PRtt PRtseapprox, [%] 0.110 0.509 0.519 0.0348 0.0405GCIfine, [%] 0.0374 0.3339 0.3421 0.0033 0.0037CPUs 240 240 240 240 240Simulations time 2h 2h 12min 2h 27min 3h 27min 4 h 18 min

55

The standard mesh is illustrated in Figs. 27-28d below.

Fig. 27: Mesh 3 in cross section.

(a) Close up of wheel domain. (b) Inlet-to-wheel fluid

(c) Wheel-to-diffuser. (d) Ported shroud.

Fig. 28: The average and standard deviation of critical parameters: Region R4

56

4.2 Surge line prediction

In order to know that the result from the simulations are trustworthy, an analysisof the compressor map against experimental data is first performed. In Figs.29a-29b below, the compressor map of efficiency as a function of mass flow isplotted. CFD data as circles, and experimental data as squares.

(a) Total-to-total (b) Total-to-static

Fig. 29: Efficiency compressor maps.

As seen in the plots, the CFD data follows the trend of the experimentaldata quite good. The same investigation where made on the compressor mapfor the pressure ratio, as seen in Figs. 30a-30b below.

57

(a) Total-to-total. (b) Total-to-static.

Fig. 30: Pressure ratio compressor maps.

Here, the experimental values are plotted as stars. Also in these compressormaps, the CFD data follows the same characteristics as the experimental data.With that knowledge, investigation of surge indicators could be performed. Asmentioned, over 140 parameters was investigated, but only parameters withindicators will be presented below. These indicators are found on the interfacesof the wheel fluid, and can be seen in Fig. 18, in section 3.2, where the examinedinterfaces are illustrated.

4.2.1 Mass flow

First out was investigation of the mass flow. By examining the wheel inletinterface (inducer), it was noticeable for the two lowest speed lines a oscillatingbehaviour at low mass flow rates, as seen in Fig. 31.

58

Fig. 31: Mass flow at wheel inlet interface

The oscillating behaviour is very small and hard to catch. Therefore a newplot was examined where the mass flow at the wheel inlet interface was plottedas a ratio over the mass flow over the compressor inlet. This is seen in Fig. 32below.

Fig. 32: Mass flow ratio, wheel inlet over compressor inlet.

By examining this plot, the behaviour is more noticeable. An indication ofsurge is spotted at the two lowest speed lines, for low mass flow rates. The

59

same mass flow ratio was also examined for the ported shroud interface over thecompressor inlet, as seen in Fig. 33

Fig. 33: Mass flow ratio, ported shroud over compressor inlet.

Once again is an indication found for the two lowest speed lines at low massflow rates.

4.2.2 Pressure recovery coefficient

In Fig. 34a and 34b below, the pressure recovery coefficient is examined at theported shroud and wheel outlet interface.

(a) Ported shroud interface. (b) Wheel outlet interface.

Fig. 34: Pressure recovery coefficient.

60

In both cases, indication of surge can be spotted for the two lowest speedlines for low mass flow rates.

4.2.3 Static pressure

From investigation of static pressure, one interface picked up a behaviour thatdiffered at the two lower speed lines at low mass flow rates. It was found byexamining the ported shroud interface. The indication was however small oscil-lations hard to see, if not knowing what to look for. Therefore, a measure planewas introduced 1 cm upstream of the ported shroud cavity, where examined. InFigs. 35a-35b, the result of these two figures are plotted.

(a) Ported shroud interface. (b) Ported shroud measure plane

Fig. 35: Static pressure at ported shroud and 1 cm upstream in ported shroud.

As seen in Fig. 35a the behaviour can be spotted for the two lowest speedlines for low mass flow rates as small oscillations. The indication is more ap-parent by investigating the upstream measure plane in Fig. 35b. Still only forthe two lowest speed lines.

4.2.4 Temperature

By placing two circumferential lines, one at a 50% radial distance from the axis,and one at a 90% radial distance from the axis, and take the mean temperatureon this lines. The ratio of these temperatures are plotted in Fig. 36 below.

61

Fig. 36: Mean temperature ratio on measure lines on wheel inlet interface.

This examination shows the most apparent indication of surge for the twolowest speed lines, for low mass flow rates.

In Fig. 37 below, the static temperature on the wheel inlet was taken as anarea average.

Fig. 37: Static temperature at wheel inlet interface.

62

This indication, besides for the two lowest speed lines also picks up thebehaviour for the highest rotational speed of 110k RPM .

The last indication of surge was found by examining the area average of thetotal temperature, once again on the wheel inlet. This is seen in Fig. 38 below.

Fig. 38: Total temperature at wheel inlet interface.

Once again, the indication of surge is only spotted at the lowest two speedlines, for low mass flows.

63

4.3 Conjugate Heat Transfer

4.3.1 Temperature fields

In Fig. 39 the temperature distribution in the compressor house (solid) is plot-ted. The temperature distribution for the compressor wheel can be seen in Figs.40a-40b.

Fig. 39: Cross section plane of temperature distribution in solids.

(a) View 1 of compressor wheel. (b) View 2 of compressor wheel.

Fig. 40: Two views to illustrate the temperature distribution on the compressorwheel.

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A comparison of the temperature on the compressor wheel was made betweenthe two cases and is presented in Table 8 below.

Table 8: Temperature distribution on compressor wheel, between CHT andadiabatic models.

Min temp, [◦C] Max temp, ◦C]CHT model 55 140Adiabatic model 25 180

In Figs. 41a and 41b below, the temperature distribution of the fluids areshown. CHT model to the left and adiabatic model to the right.

(a) Cross section view of temperature dis-tribution on CHT model.

(b) Cross section view of temperature dis-tribution on adiabatic model.

Fig. 41: Comparison of temperature distribution, CHT model to the left, adia-batic model to the right.

Another comparison of temperature distribution in the fluid is made onceagain, but here for the diffuser. CHT model to the left and adiabatic to theright, as seen in Figs. 42a and 42b below.

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(a) Cross section view of temperature dis-tribution in diffuser on CHT model.

(b) Cross section view of temperature dis-tribution in diffuser on adiabatic model.

Fig. 42: Comparison of temperature distribution, CHT model to the left, adia-batic model to the right.

4.3.2 Efficiency

Next up is numerical plots for comparison of the different models. In Figs.43a and 43b below, the efficiency is plotted together with adiabatic traditionalmodel as well as experimental data for comparison.

(a) Efficiency, total-to-total. (b) Efficiency, total-to-static.

Fig. 43: Efficiency comparison between CHT model, traditional adiabatic modeland experimental data.

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4.3.3 Pressure Ratio

The pressure ratio of the CHT model is also compared against the traditionaladiabatic model and the experimental data, as seen in Figs. 44a and 44b below.

(a) Pressure ratio, total-to-total. (b) Pressure ratio, total-to-static.

Fig. 44: Pressure ratio comparison between CHT model, traditional adiabaticmodel and experimental data.

In figure 45 below, the power of the compressor wheel is compared betweenthe two models.

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Fig. 45: Difference in power between the CHT and adiabatic model.

4.3.4 Temperature

In the next following plots, there is no experimental data to compare against, sothese plots are only to compare the CHT model against the traditional adiabaticmodel. First is the total temperature ratio and total temperature at wheeloutlet, as seen in Figs. 46a and 46b below.

(a) Total temperature ratio. (b) Total temperature at wheel outlet.

Fig. 46: Total temperature comparison.

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In Figs. 47a and 47b below, the total temperature at the ported shroud andthe wheel inlet is plotted.

(a) Total temperature at ported shroud. (b) Total temperature at wheel inlet.

Fig. 47: Total temperature comparison.

4.3.5 Static pressure

The static pressure at the outlet and ported shroud is seen in Figs. 48a and 35abelow,

(a) Static pressure at outlet. (b) Static pressure at ported shroud.

Fig. 48: Static pressure comparison.

and the same parameter measured at the wheel inlet and wheel outlet is

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plotted in Figs. 35a and 49a below.

(a) Static pressure at wheel inlet. (b) Static pressure at wheel outlet.

Fig. 49: Static pressure comparison.

4.3.6 Total pressure

The last results is for the total pressure, where the parameter is measured atthe outlet and wheel outlet can be seen in Figs. 50a and 50b below.

(a) Total pressure at outlet. (b) Total pressure at wheel outlet.

Fig. 50: Total pressure comparison.

The same parameter measured over the ported shroud and wheel inlet isplotted in Figs. 51a and 51b below.

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(a) Total pressure at ported shroud. (b) Total pressure at wheel inlet.

Fig. 51: Total pressure comparison.

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5 Analysis & Discussion

5.1 Mesh dependency study

From Fig. 25a, 25b and 26 the examined parameters is plotted as a function ofmesh refinement. In all three figures, the converging behaviour of the functionvalues is found. This converging behaviour is sought, and can graphically tellhow the function value eventually will not change anymore.

The approximate relative error for the examined parameters between Mesh3

and Mesh4 are less than 0.52% and the numerical uncertainty for the sameparameters are less than 0.35%.

The added simulation time to perform computations on Mesh4 is approxi-mately one hour longer than for Mesh3. This is only the time for computingthe case, and the time for discretization is not included. That process increasedin man-hours for every refinement.

As mentioned earlier, the CFX manual in Ansys recommends to use at leastl0 layers or more when using inflation methods, to properly resolve the boundarylayer.

All above points has been used to motivate the use of Mesh3 for futurework in this thesis project, since it is refined enough. As seen in Table 4,Mesh3 consisted of 37, 915, 257 elements.

5.2 Surge line prediction

From the performed steady state simulations, indicators of the surge phenomenawas found. These indicators where often found by examining wheel inlet andported shroud. This did not come as a surprise since surge arises because of largesystem oscillations or flow instabilities due to the large pressure gradients in theregion around the compressor wheel. With other words, the compressor wheelis the region in the compressor that should be affected by the phenomena, andfor all indicators found, they were found in the vicinity of the compressor wheel.

From the plots that could indicate of surge, it could be traced specifically toa certain region in the mass flow for the two lower speed lines. Because of theconfidentiality of this thesis, no numerical value of this mass flow regions willbe presented.

The higher rotational speeds simulated for, the harder was it to capture thisphenomena. In fact only one indication of surge was found. Static temperatureat wheel inlet at 110k RPM . Because only one indication was found, no con-clusion can be drawn that surge will appear at this mass flow. Therefore moreindications are needed to ensure the mass flow region of surge of higher speedlines. The same indicator was not found for the speed line of 87k RPM . Byexamining Fig. 37, it almost looks like the design points simulated over has notreach the surge region yet and needs more design points.

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It can also be answered by the fact that surge is both a stationary andtransient behaviour, and might need transient simulations to indicate on surgefor higher speed lines.

5.2.1 Future work

Since it in many plots looks like the design points for higher speed lines arenot in the region of surge, it might mean that more design points are neededin the region of surge to capture the behaviour. The design points was takenfrom experimental data, so it should be checked that the experimental part wasperformed correct for all speed lines.

Because the simulations could not capture behaviour of surge at higher speedlines using steady state simulations, the next step would to perform this studywith transient simulations.

Surge is a phenomena that not only depends on the compressor but the thesystem as a whole. The inlet was extruded 8 diameters to ensure a stable flowcomputed over. It would have been interesting to compare this extruded inletwith an original inlet, but also against other inlet types of compressors, such asradial inlets. To see if this geometrical change makes it easier to predict surge.For that reason, it would also be of interest to compare two different compres-sors with different size and volume, to see if volumetric or geometric differencescan effect the ability to indicate surge.

5.3 Conjugate Heat Transfer

The geometry for the CHT model as stated in section 3.3 should nominallybe the same as for the adiabatic model. When all simulations were performedand the results where analyzed the geometry was compared to justify the result.From Table 6, it is noticeable that there are several distances that differ betweenthe two geometries. Especially hps, the height of the ported shroud interface.The height of the CHT model is almost half the height of the adiabatic model,which makes it harder for the gas to flow through the ported shroud cavity.This also means that when comparing results from the CHT model against theadiabatic model and experimental data, the geometry is not the same. It hasbeen shown from earlier work at Scania that geometrical differences in the re-gions in vincinity of the compressor wheel can have great affect on the results.Therefore one has to be reminded of this when analysing the results comparingthe different cases.

As seen in Fig. 39, the temperature distribution in the solids, (compressorhouse) can be solved for. The same goes for the heat transfer on the compressorwheel and shaft, as seen in Figs. 40a and 40b. This is one of the benefits ofperforming a CHT model, compared to using adiabatic outer walls of the fluid,

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as usually is done at Scania. These temperature fields, but also wall heat flux,is vector fields that can be exported from CFD-Post to be used for mechanicalanalysis.

The lowest temperature of the CHT model is more than double the temper-ature of the adiabatic model as seen in Table 8. It also differ at the maximumtemperature, where the CHT model is 40◦C lower. This might depend on thefact that the CHT model allows heat transport between fluids and solids. There-fore the gas will pick up heat from the solids on its way down to the compressorwheel, leading to a higher minimum temperature on the wheel. With the samereason, the compressor wheel will emit heat to the fluid and surrounding solids,leading to a decrease in maximum temperature.

In Figs. 41a and 41b the temperature distribution in the fluid is presented.It its clearly noticeable the quantitative difference in how the CHT model (left)handles the heat transport compared to the adiabatic model (right). This getsmore obvious by also examining the temperature distribution in the diffuser fluiddomain, Figs. 42a and 42b. There is a quantitative difference in the distributionof temperature once again. Worth to mention is that the temperature scales forthese plots are the same when comparing the CHT against the adiabatic model.

When comparing the efficiency in Figs. 43a and 43b, it is noticeable thatthe CHT model follows the same trend as the traditional adiabatic model, butalso as experimental data. However when approaching lower mass flows, theCHT model starts to underpredict in function value. This underprediction ofefficiency could have to do with the fact that the CHT model allows for heattransfer out of the fluid walls, whilst the adiabatic model does not, and is con-sidered a isolated system that does not allow heat or matter transport throughwalls. Therefore the CHT model was also tested without the solid domains, andwith adiabatic outer walls of the fluid domains. This CHT model with adiabaticwalls were performed at one design point of 0.1671 kg/s in mass flow, and isplotted as the red star in Figs. 43a and 43b. Corresponding design point withCHT model is plotted as a blue star. Unfortunately the efficiency dropped evenmore as seen in the figures, which lead to the conclusion that the allowance ofheat transfer in the CHT model is not the reason for the efficiency to underpre-dict.

Instead an analysis of the pressure ratio was performed, as seen in Figs.44a and 44b. Once again the CHT model follows the same characteristics asthe adiabatic model and experimental data, but starts to underpredict whenapproaching lower mass flow rates. Because of that, the power in the systemwas compared between the CHT and adiabatic model. The allowance of heattransfer in the CHT model might lead to a loss of energy in the system thatcould be traced back to the underprediction of efficiency and pressure ratio. Asseen in Fig. 45, the power for the CHT model and the adiabatic model has thesame curve and is only shifted in function value. If the energy loss would have

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been the reason for the under prediction, the power for the CHT model shouldthen have a deviating behaviour for lower mass flows compared to the adiabaticmodel. This was not the case, which lead to the conclusion that the underpre-diction of efficiency and pressure ratio of the CHT model rather depends on thegeometrical differences shown earlier in section 3.3, in Fig. 22, and discussedabove.

The total temperature ratio between outlet and inlet as seen in Fig. 46ashows a quantitative difference in function value, but both models have a simi-lar behaviour.

From Fig. 46b, showing the total temperature at the wheel outlet, the twomodels behave relatively the same with little deviation.

When analyzing the total temperature at the ported shroud and wheel inlet,seen in Figs. 47a and 47b the CHT model have a whole other behaviour thanthe adiabatic model for lower mass flows. For increasing mass flow the CHTmodel starts to follow the characteristics of the adiabatic model. The reason forthe strange behaviour at low mass flows may depend on the usage of the portedshroud. The ported shroud is used to recirculate the flow when the compressoris performing under low mass flows. In the CHT model the gas will from theheat transfer in the system pick up heat energy leading to this behaviour. Whenthe mass flow is increased, the ported shroud is not needed since the gas natu-rally can flow downstream of the compressor without stalling. This is why thebehaviour of the CHT model starts to follow the trend of the adiabatic modelfor increasing mass flow rates.

The static pressure in Fig. 48a and 35a both shows a similar behaviour.The pressure on the ported shroud is however much higher in the CHT model.This can be explained by the interface height of the ported shroud in the CHTgeometry that is almost the half of the height of the adiabatic geometry. Thesmaller opening of the interface will produce an increase in pressure over theinterface.

By comparing the static pressure at the wheel inlet and wheel outlet seenin Figs. 49a and 49b, the pressure is similar at the wheel inlet. In the CHTmodel, the static pressure at the outlet is however increased, meaning that theCHT model computes a pressure ratio over the compressor wheel that is highercompared to the adiabatic model.

The total pressure at the outlet is following the same characteristic betweenthe models, as seen in Fig. 50a. In Fig. 50b, the total pressure at the wheeloutlet shows a qualitative difference, but follows the same trend.

The behavior at the ported shroud is once again deviating for lower massflows, but straightens out when the flow increases. This is seen in Fig. 51a and

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can also be justified by the geometrical differences at the ported shroud interface.

At the wheel inlet, the total pressure looks to differ a lot. This is not trueand can be explained by the normalized plot that gives a bad scale. The func-tion values of the total pressure at the wheel inlet is in fact fairly close betweenthe two models. This means, by looking at Figs. 50b and 51b that the totalpressure ratio over the compressor wheel is lower for the CHT model.

Overall based on efficiency and pressure ratio, the CHT model shows a char-acteristic fairly close to the adiabatic model and experimental data but un-derpredicts when approaching lower mass flows. These underpredictions maydepend on the geometrical differences.

5.3.1 Future work

When conducting CHT simulations including MRF, there are recommendationsof letting every MRF solid domain be encased by a fluid domain that should berotating along with the solid at the same rotational velocity. From earlier workat Scania, it has been found out that the positioning of the wheel domain inter-faces have a significant effect on parameters such as efficiency. For that reason,the wheel fluid domain is starting at the same axial distance as the traditionalmodel is, and the same goes for the wheel outlet to diffuser interface. A problemwith this approach is that for the CHT model, a shaft and shaftnut domain isalso included in the system. Both of these domains are starting inside the inletfluid domain, which is set as a stationary domain. Since the shaft and shaftnut is rotating with the same rotational velocity as the compressor wheel, thosedomains should also be encased by a rotating fluid, which is not the case. If thishas a negative influence on parameters such as efficiency would be interestingto investigate.

The time scale used in the solver is based on recommendations from a Ansyswebinar in the subject of CHT, that states that an automatic time scale shouldbe used [6]. In the finishing parts of this thesis, i found an article about CHTmodeling, stating that the solid time scale should be set up using the equation:

Ts =Lsρ, Cp,s

λs(46)

where s denotes the solid. Here L is a characteristic length of the solid, andλs the thermal conductivity of the material. Since the CHT simulations is per-formed with the automatic time scale factors for the fluid and solid of 10 and100 respectively, it would be of interest to see the difference in the result byusing the traditional physical time scale for the fluid, together with eq. (46) forthe solid.

It would also be of interest to perform a mesh study, both on fluid and soliddomains in the geometry. Especially for the solids, so the assumption of using

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a coarse mesh because of the linearity of the heat transfer in solids, can be tested.

The CHT model was also performed with adiabatic walls, for one designpoint. Because of the time limit, I had no time to perform this CHT modelwith adiabatic walls for the complete speed line, which would have been inter-esting to do. Firstly because it should behave more like the adiabatic traditionalmodel. Secondly because the CHT model had geometrical differences and couldtherefore not fully be compared to the adiabatic model, since the differences inthe result is dependent on the geometry.

To perform a transient simulation to compare against adiabatic model andexperimental data would also be of interest, to see if a time dependent solutionshave better prediction.

Finally, to perform a more comprehensive experimental study with more datafor evaluation of the CHT model to better understand if it is a good model.

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[12] John W. Slater. Examining Spatial (Grid) Convergence. url: https://www.grc.nasa.gov/WWW/wind/valid/tutorial/spatconv.html.

[13] Elias Sundstrom. “Flow instabilities in centrifugal compressors at lowmass flow rate”. 2017, pp. 32–35. isbn: 978-91-7729-555-6.

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